wanminliu@gmail.com
2021
0917
l
If
you
are
using
a
computer
,
you
may
open
sage.math.org
and
click
locale
Instant
Sage
Worksheet
wanminliu@gmail.com
Sagemath
§ly¥|
Find
Taylor
series
of
=
#
¥
f-
ix.
7)
=
z¥zy-
hear
12.11
ft=s_⑤~
+
is
uptodeg3→
near
o
.
¥
let
5=11-2
÷*=±É
"
'
"
w
-1
for
/
×
,
,
⇐
hear
10,0
)
Ktm
wrote
it
up
to
n
=
,
hear
12.1
)
fr
is
.ws
=
If
it
lw-s-d-tw-sj-ya.gg
put
back
we
y
-1
f-
IX.
4)
=
1-
six
-
e
2+4+4
-2kW
-111
f%ame×r%É+ tÉg
=
z¥Ñ
wanminliu@gmail.com
https://www.sagemath.org/ and its online version https://cocalc.com/
Documents: https://doc.sagemath.org/html/en/prep/Quickstarts/Multivariable-Calculus.html
wanminliu@gmail.com
551¥
.
IX.
Y
.
7)
We
WANT
To
Find
'
*
maxi
•
*
.
with
the
constrain
¥¥='
Given
×a÷
+
¥
+
¥
=
1
,
✗
70. 770
.
2-30
Fintmethod
.
standard
Then
Z
=
2-
(
X
-
Y
)
Voll
box
)=
81×72-1
put
it
into
.
8×72-1×-41
We
can
pick
up
IX.
Y
,
Z
)
with
✗
so
Find
Witial
point
g-
of
y
so
the
two
vakwk efrm.lu
>
o
2ndmethad_
Lag
.
multiplier
.
wanminliu@gmail.com
§l4
extreme
val ues
of
function
on
a
restricted
domain
e-
Rem
requirement
.
•
closed
•
bounded
]
existence
of
ext rem e
values
(
maximin
'll
chelk_
.
interior
point
(
same
as
914.1
)
critical
point
vector
8=10
.
.
-
;D
if
I
c-
Domlf
)
and
Of
1×1=8
,
then
I
is
a
critical
point
.
Tfi
)
=
(
f
,
,
.
-
i.tn
)
•
singular
point
I
☐
f-
1¥
has
no
def
.
)
New
in
this
satin
•
Boundary
points
.
wanminliu@gmail.com
f-
10.07=0
.
%
There
is
no
singular
point
.
=
Find
the
mail
&
"
"
"
•
t.me#mdaypo-mtsIflX.y)--2XY
on
the
closed
disk
✗
'
+554
③
✗
4-
y
'
=
4
.
#
let
✗
=
260
-
aeo .cz
•
Dom
(f)
=
{
IX.
yj
/
✗
4-7%47
y
=
2
Sino
is
bounded
&
closed
.
gloj-ttflx.vn/--f(zww.2s2O1--8sinows o
boundary
•
f-
is
continuous
.
Th
2
f
has
absolute
mail
dminlNFFgindtw-w.ae
values
at
some
point
g-
the
disk
.
problems
on
a
rester
.
domain
to
a
one
variable
problem
.
①
Let
us
find
critical
points
,
,
Find
max
'll
mind
of
9101
.
☐
f-
=
4-
×
,
f-
y
)
=
127
,
2x
)
9101=4.12
Sino
Wil
)
=
4
.
sin
20
.
go
ryf
(
x
.y
,
=o
(
✗
/
7)
=
(
°
.
°
)
-271<-20
£22
5Th
10
zo=¥
,
¥
-
za
20=-7-2
,
+27
go
[
o.o
,
is
the
only
attend
point
and
Éma×Q
fix
,y)=
4
.
min
=
-4
①
②
③
conclusion
:
\
wanminliu@gmail.com
f-
10.07=0
.
#
3
f-
IX.
y
)
=
✗
y
-
yz
⑦
There
is
no
singular
point
.
Find
the
mail
•
"
"
"
t.me#mdaypo-mts&
-
③
on
the
closed
disk
✗
'
+5ft
✗
4-
y
'
=p
Sol
let
✗
=
cost
-
aeo .cz
•
Dom
(f)
=
{
(
Ky
)
/
✗
4-
YEP
}
y
=
Sino
is
bounded
&
closed
.
gloj-tflx.vn/--f(ww.s20l--cos osino--siio
boundary
•
f-
is
continuous
.
Th
2
f
has
absolute
mail
dminlNFFrigindtw-w.ae
values
at
some
point
g-
the
disk
.
problems
on
a
rester
.
domain
to
a
one
variable
problem
.
①
Let
us
find
witted
points
)
.
Find
max
-1
/
mind
of
9101
.
Of
=
Ctx
,
f-
y
)
=
(
Y
,
X
-
4)
So
f-
IX.
y
,
=o
(
✗
it
)=(
0,0
)
so
0.0
)
is
the
only
attend
point
and
①
⑦
③
wanminliu@gmail.com
gloj-ttflx.vn/--flww.s201--cososino--siio
2STn0wd=fT%
boundary
1-
25in
'll
=
v20
→
EOE
-17
Dont
g)
=
bounded
d
closed
.
-
We
solve
the
min
mail
pwblemtr
g
g'-
9401=-81
'nÑ
+
two
sooo
-
zsinoae
=
Uzo
-
5m20
g'
(D)
=o
-4
cozO=shn2O
Find
crit
.
point
of
0
in
the
domain
.
__
✗
=
20
I
/
F-
¥
swamp
+
was
'mP
910
)=
t
sinzo
+
Icona
-
I
=
Sink
-1ps
=
⑤
since
-1¥
wzo
)
-
I
sin
120
-1¥
)
-
I
50-1
C-
C-
¥-1
,
+
Fi
-
£
]
m
:n=
-
¥-1
,
Max
=
-
I
☐
wanminliu@gmail.com
814.3
Lagrange
multiplier
Th4_
.
Suppose
f-
fix
.y
)
&
g-igix.yjsatsfy.mg
Then
7-
to
C-
IR
S.t.
•
¥
.
¥
.
1¥
.
¥y
continuous
.
(
Xo
.
Yo
,
to
)
is
a
critical
point
near
Pot
Ho
-
%)
4-
Devote
c-
=
{
ix.
x
)
/
gix
.y)=o
}
•
(
↳
r-oetmti.ua
"
"
so
it's
a
curve
ratable
>
)
•
suphalflx
-
y
)
/
e
has
a
local
maximin
:p
L(
✗
i.
b)
¥
ftx.vn/-tgcx.y
at
Po
Then
along
•
suppose
that
%
Po
is
NOT
an
endpzy.ge#ipT9Po--
÷÷"¥¥÷m
change
lid
V91
B)
1=8--10,0
)
if
yep
,
*
pg
,µwme=%
'
is
NO T
led
we
could
alury
pwj
.
fcpo
)
to
1¥
9
(
B
)
)
"-
wanminliu@gmail.com
-1-13-1
find
the
distance
from
the
Yin
①
she
H
is
on
the
plane
ttheplanex-ZY-zz-3.ge
6
lab
.
c)
satisfy
(a)
Using
a
geometric
argument
.
At
2b
-12
2-
=3
.
(A)
•
0=10.0
,
0
)
0+20+20=0
①
0¥
is
a
normal
vector
of
so
①
is
not
on
the
plane
.
We
dlt
.
H
the
plane
.
g-
S.t.
OH
1-
Plane
at
✗
(
1.
2.
2)
Tefft
linear
terms
.
y¥É
9-
=
¥
=
I
(B)
a-
§
b=
}
.
[
=
§
dis
1-10
,
plane
)
=
dhtfo.t l/bewteH=la.s
.
c)
dist
=
dbtlo.tl/=#+ 2-i+fgj--
I
☐
wanminliu@gmail.com
(b)
by
reducing
the
problem
to
an
☐
f-
IX.
y
)
unconstrained
problem
in
two
variables
.
=
(2×+141×+24-3)
,
24+3=1×+24-33-4
dist
'=
✗
4742-2
8th
.
-11=0
with
constrain
✗
-124+22=3
⇐
/
Ex
-17-23=0
✗
+47-3=0
/
✗
=
}
2-
=
-113
-
x
-
27
)
y=¥
dat
'=
✗
'
+5+-412×-45
min
dirt
'
=
distills
.
}
)
for
two
variables
✗
dy
=
I
Find
mind
.
f-
distill
.
plane
)
)
dttlo
.
Plane
)
=L
wanminliu@gmail.com
④
Lag
.
multiplier
.
94.4
.
2-
I
=
✗
+
Ht
22--3
gives
the
minYma
L
IX.
4.
a.
d)
=
f-
+
✗
g
⇐
÷
:*
,
"
"
"
"
"
"
"
"
"
fit
,
}
,
;-)
=
B.
Them
-
TL
=
(
Lx
.
Lxy
.
Lz
4)
=
(
2×14.4+24,22-+4
,
knyaz
,
,
µ"kk€3EPlaY
debt
10.01=3
>
1
.
let
☐
[
=
.
{
linen
system
}
-
So
this
is
the
mini
↳
dirt
☐
wanminliu@gmail.com
(d)
Cauchy
-
Schwartz
ineg
.
/
of
.
I
/
E
/
Ñ
/
II
/
Ie
.
Kai
:
an
)
•
lbii.by/EaiF-aibi--bnT
"
tettheplanebeA-x -By-cz-p .no/AH+''-+a*#b-+-
a
:*
,÷
.
+
big
for
some
numbers
A
,
B
,
C
and
D
.
let
P
=
ix.
Y
.
7)
be
apo-mtontheplme-sofn-x-By-cz-J-illtpo-lxo.7.to
)
be
a
point
outside
the
plane
.
-
dist
'
(
Po
.
P
)
=
Ix
-
Xo
)
7-
4-
%)
7-
¢-25
(
A'
+
B'
+
e)
(
ex
-
✗
→
7-
e-
%
>
'
+
(2--2-5)
A
1×-161+13174>+42--2*1
'
=/
A
Xo
-113%-1
(
Zo
+
D
/
2
¥
☒
-
x.it#Ysi-st-a5=/AxaI??;?Y-Dl---diitcB.p1ane
,
☐
'
wanminliu@gmail.com
-
hhythsisthemin@Hn-venteatt ll_dht.lO
,
Snfae
)
0-441.11
TL
=
0
dirt
1001
>
f
SWf.ae
:
✗
y
'
2-
✗
=
32
.
2×+2722-4=0
①
27+24×72-4=0
②
¥
(
↳
+
⇐
✗
yz×=o
③
bet
fix
-4.2-1=1/45+2-2
.
✗
y
-2-4=32
④
(
dirt
.
square
function
)
min
flx.Y.tl
2×1211--24-7=2-2
②
✗
Y
with
constrain
gcx
.%z)=×y2z✗→z=o
§
=
-
let
Lucy
#
d)
=
flx.y.tl/-t9lX.Y.t1 V--i-zxi-4 X'/x--i7L--ffx-X9x .fy-xsy,fz-tgz
,
9)
=
>
dit
=D
.
☐
wanminliu@gmail.com
-1-115-1
max
'd
/
min
't
flx.y.tk
4
-
z
subject
to
9
,
ix.
Y
.z
,
=
✗
I
-1^-8=0
f%Ac.Y.tl-x-y-t-1-osam-meth-al.by
we
introduce
two
new
cariole
,
d.
M
[
IX.
7.7
,
HM
)=
f-
+79
,
-1M
92
=
fcx.az
)
-1×9
,
IX.
Y.tl/-jh9zlx.Y.Z1
•
calculate
0L
=
(
Lx
.
↳
.
Iz
,
Li
.
Lu
)
•
Find
critical
points
,
0L
=
8
•
solve
it
.
set
utkal
point
•
plug
them
into
f-
IX.
7.
7)
•
tent
-
Maxie
\
min
't
.
wanminliu@gmail.com
814.4-1
n
IX.
5-
1=0
-11-1-1
Find
the
maximin
val ues
of
xi-nn-var.fm
Hi
,
.
.
-
.in/--lfrnj!--.FnJjf-lX,.i-,Xn)--Xi-.-.-XnswJ.to.tle
constrain
✗
ix.
.
.+×÷
,
.
"
IX.
'
-
XD
__
tf
?
-
-
-
;
-
frnjj
-
so
plug
them
into
f.
¥
LIX
.
.
.
.vn
,
d)
=f
-179
fern
,
-
-
rn
)=n¥=rn
Max
where
glx
.
.
.vn
)=
✗
it
.
-
.
-1×5-1
fi-rn.i-rn-n-fn.vn
.
PL
=\
1+21×1
,
'
-
.
.
ltzxn
,
✗
it
.
.in?
,
)
ohh
.
→
☐
L=o
✗
,
=
-
¥
=
.
.
=✗
;
